Pairs of Linear Equations: Quick Revision

A brief note on Pairs of Linear Equations. Read it before a test. Try each question at the end before you open its answer.

Three cases

A solution of a pair is a point on both lines. Write the lines as $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$.

• $\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$: the lines: cross; solutions: one
• $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}$: the lines: parallel; solutions: none
• All three equal: the lines: the same line; solutions: infinitely many

Three ways to solve

• Graph: draw both lines and read the crossing. Good for seeing, not always exact.
• Substitution: write one variable in terms of the other and put it into the second equation. Keep the brackets.
• Elimination: add or subtract the equations, after multiplying if needed, so that one variable cancels.

A false line such as $0 = 5$ means no solution. A true line such as $0 = 0$ means infinitely many.

Problems in words

Name the two unknowns, write one equation for each fact, solve, and check the answer in the words.

A worked example

Solve $3x + 2y = 20$ and $x + 2y = 8$.

1. Subtract. The $2y$ terms cancel: $2x = 12$, so $x = 6$.
2. Back substitute. $6 + 2y = 8$, so $y = 1$.
3. Check. $18 + 2 = 20$ and $6 + 2 = 8$.

Practice with new numbers

A program to try

The program works out the worked example.

Mistakes to avoid

• Checking the answer in one equation only. It must fit both.
• Dropping the brackets in a substitution. $2(y + 1)$ is $2y + 2$, not $2y + 1$.
• Reading $0 = 0$ as $x = 0$. It means the two equations are the same line.
• Answering for the wrong unknown. In a boat problem, say whether you found the boat or the stream.

Check yourself

1. How many solutions has the pair $x - 2y = 3$ and $3x - 6y = 9$?

2. Solve $x + y = 16$ and $x - y = 6$.

3. Solving a pair ends with $0 = 7$. What does that mean?

4. Two pens and a pencil cost $9$ dollars. One pen and a pencil cost $6$ dollars. What does a pen cost?